Process models that are utilized for prediction, control and optimization can be divided into two general categories, steady-state models and dynamic models. In each case the model is a mathematical construct that characterizes the process, and process measurements are utilized to parameterize or fit the model so that it replicates the behavior of the process. The mathematical model can then be implemented in a simulator for prediction or inverted by an optimization algorithm for control or optimization.
Steady-state or static models are utilized in modern process control systems that usually store a great deal of data, this data typically containing steady-state information at many different operating conditions. The steady-state information is utilized to train a non-linear model wherein the process input variables are represented by the vector U that is processed through the model to output the dependent variable Y. The non-linear model is a steady-state phenomenological or empirical model developed utilizing several ordered pairs (Ui, Yi) of data from different measured steady states. If a model is represented as:Y=P(U,Y)  (1)where P is some parameterization, then the steady-state modeling procedure can be presented as:({right arrow over (U)},{right arrow over (Y)})→P  (2)where U and Y are vectors containing the Ui, Yi ordered pair elements. Given the model P, then the steady-state process gain can be calculated as:
                    K        =                              Δ            ⁢                                                  ⁢                          P              ⁡                              (                                  U                  ,                  Y                                )                                                          Δ            ⁢                                                  ⁢            U                                              (        3        )            
The steady-state model therefore represents the process measurements that are taken when the system is in a “static” mode. These measurements do not account for the perturbations that exist when changing from one steady-state condition to another steady-state condition. This is referred to as the dynamic part of a model.
A dynamic model is typically a linear model and is obtained from process measurements which are not steady-state measurements; rather, these are the data obtained when the process is moved from one steady-state condition to another steady-state condition. This procedure is where a process input or manipulated variable u(t) is input to a process with a process output or controlled variable y(t) being output and measured. Again, ordered pairs of measured data (u(I), y(I)) can be utilized to parameterize a phenomenological or empirical model, this time the data coming from non-steady-state operation. The dynamic model is represented as:y(t)=p(u(t),y(t))  (4)where p is some parameterization. Then the dynamic modeling procedure can be represented as:({right arrow over (u)},{right arrow over (y)})→p  (5)
Where u and y are vectors containing the (u(I),y(I)) ordered pair elements. Given the model p, then the steady-state gain of a dynamic model can be calculated as:
                    k        =                              Δ            ⁢                                                  ⁢                          p              ⁡                              (                                  u                  ,                  y                                )                                                          Δ            ⁢                                                  ⁢            u                                              (        6        )            
Unfortunately, almost always the dynamic gain k does not equal the steady-state gain K, since the steady-state gain is modeled on a much larger set of data, whereas the dynamic gain is defined around a set of operating conditions wherein an existing set of operating conditions are mildly perturbed. This results in a shortage of sufficient non-linear information in the dynamic data set in which non-linear information is contained within the static model. Therefore, the gain of the system may not be adequately modeled for an existing set of steady-state operating conditions. Thus, when considering two independent models, one for the steady-state model and one for the dynamic model, there is a mis-match between the gains of the two models when used for prediction, control and optimization. The reason for this mis-match are that the steady-state model is non-linear and the dynamic model is linear, such that the gain of the steady-state model changes depending on the process operating point, with the gain of the linear model being fixed. Also, the data utilized to parameterize the dynamic model do not represent the complete operating range of the process, i.e., the dynamic data is only valid in a narrow region. Further, the dynamic model represents the acceleration properties of the process (like inertia) whereas the steady-state model represents the tradeoffs that determine the process final resting value (similar to the tradeoff between gravity and drag that determines terminal velocity in free fall).
One technique for combining non-linear static models and linear dynamic models is referred to as the Hammerstein model. The Hammerstein model is basically an input-output representation that is decomposed into two coupled parts. This utilizes a set of intermediate variables that are determined by the static models which are then utilized to construct the dynamic model. These two models are not independent and are relatively complex to create.